At first sight finitely generated abelian groups and canonical forms of matrices appear to
have little in common. However reduction to Smith normal form named after its originator
H.J.S.Smith in 1861 is a matrix version of the Euclidean algorithm and is exactly what the
theory requires in both cases. Starting with matrices over the integers Part 1 of this book
provides a measured introduction to such groups: two finitely generated abelian groups are
isomorphic if and only if their invariant factor sequences are identical. The analogous theory
of matrix similarity over a field is then developed in Part 2 starting with matrices having
polynomial entries: two matrices over a field are similar if and only if their rational
canonical forms are equal. Under certain conditions each matrix is similar to a diagonal or
nearly diagonal matrix namely its Jordan form. The reader is assumed to be familiar with the
elementary properties of rings and fields. Also a knowledge of abstract linear algebra
including vector spaces linear mappings matrices bases and dimension is essential although
much of the theory is covered in the text but from a more general standpoint: the role of
vector spaces is widened to modules over commutative rings. Based on a lecture course taught by
the author for nearly thirty years the book emphasises algorithmic techniques and features
numerous worked examples and exercises with solutions. The early chapters form an ideal second
course in algebra for second and third year undergraduates. The later chapters which cover
closely related topics e.g. field extensions endomorphism rings automorphism groups and
variants of the canonical forms will appeal to more advanced students. The book is a bridge
between linear and abstract algebra.
